A271535 - OEIS

%I #56 Feb 25 2023 06:34:37

%S 0,1,25,196,900,3025,8281,19600,41616,81225,148225,256036,422500,

%T 670761,1030225,1537600,2238016,3186225,4447881,6100900,8236900,

%U 10962721,14402025,18696976,24010000,30525625,38452401,48024900,59505796,73188025,89397025,108493056

%N a(n) = ( n*(n + 1)*(2*n + 1)/6 )^2.

%H R. D. Carmichael and T. L. DeLand, <a href="http://www.jstor.org/stable/2967699?seq=1#fndtn-page_scan_tab_contents">Find the sum of the series 1^2 + 5^2 + 14^2 + 30^2 + ... + [n*(n+1)*(2*n+1)/6]^2</a>, American Mathematical Monthly, Vol. 15, No. 6/7, Jun-Jul, 1908, pp. 132-133.

%H <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (7,-21,35,-35,21,-7,1).

%F G.f.: x*(1 + 18*x + 42*x^2 + 18*x^3 + x^4)/(1 - x)^7.

%F E.g.f.: x*(36 + 414*x + 744*x^2 + 393*x^3 + 72*x^4 + 4*x^5)*exp(x)/36. - _Ilya Gutkovskiy_, Apr 21 2016

%F a(n) = 7*a(n-1) - 21*a(n-2) + 35*a(n-3) - 35*a(n-4) + 21*a(n-5) - 7*a(n-6) + a(n-7).

%F Sum_{i = 0..n} a(i) = n*(n + 1)*(n + 2)*(2*n + 1)*(2*n + 3)*(5*n^2 + 10*n - 1)/1260. [See Carmichael - DeLand in Links section, page 132.]

%F a(n) = A000330(n)^2. - _Ray Chandler_, Apr 21 2016

%F Sum_{n>=1} 1/a(n) = 84*Pi^2 - 828. - _Amiram Eldar_, Feb 25 2023

%t Table[(n (n + 1) (2 n + 1)/6)^2, {n, 0, 50}]

%o (Magma) [(n*(n+1)*(2*n+1)/6)^2: n in [0..50]];

%o (PARI) vector(100, n, n--; (n*(n + 1)*(2*n + 1)/6)^2) \\ _Altug Alkan_, Apr 21 2016

%Y Cf. A000290, A000330, A000537, A005563, A046092, A059977.

%K nonn,easy

%O 0,3

%A _Vincenzo Librandi_, Apr 20 2016

%E Edited by _Bruno Berselli_, Apr 22 2016